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The given problem is about the total weekly cost incurred by Lincoln Records for pressing \( x \) compact discs, represented by the cost function:

\[
C(x) = 2000 + 2x - 0.0001x^2 \quad \text{where} \quad 0 \leq x \leq 6000
\]

### Part (a): 
We need to find the **actual cost incurred** for producing the 981st and 1801st discs. This means we are interested in the **marginal cost** of producing the 981st and 1801st disc. The marginal cost \( MC(x) \) is found by differentiating \( C(x) \) with respect to \( x \).

#### Step 1: Differentiate the cost function \( C(x) \)

\[
MC(x) = \frac{dC(x)}{dx} = 2 - 0.0002x
\]

#### Step 2: Calculate the marginal cost for \( x = 981 \) and \( x = 1801 \).

1. **For the 981st disc**:
   \[
   MC(981) = 2 - 0.0002 \times 981 = 2 - 0.1962 = 1.8038 \, \text{dollars}
   \]
   Rounded to the nearest cent:
   \[
   MC(981) = 1.80 \, \text{dollars}
   \]

2. **For the 1801st disc**:
   \[
   MC(1801) = 2 - 0.0002 \times 1801 = 2 - 0.3602 = 1.6398 \, \text{dollars}
   \]
   Rounded to the nearest cent:
   \[
   MC(1801) = 1.64 \, \text{dollars}
   \]

### Part (b): 
Now, we calculate the marginal cost at \( x = 980 \) and \( x = 1800 \).

1. **For \( x = 980 \)**:
   \[
   MC(980) = 2 - 0.0002 \times 980 = 2 - 0.196 = 1.804 \, \text{dollars}
   \]
   Rounded to the nearest cent:
   \[
   MC(980) = 1.80 \, \text{dollars}
   \]

2. **For \( x = 1800 \)**:
   \[
   MC(1800) = 2 - 0.0002 \times 1800 = 2 - 0.36 = 1.64 \, \text{dollars}
   \]

---

Would you like more details on the differentiation process or have any questions? Here are five related questions to expand on this problem:

1. How does the marginal cost change as the number of discs increases?
2. What is the significance of the second derivative of the cost function?
3. How would the cost function change if the fixed cost of 2000 was increased?
4. What would the marginal cost be if \( x = 0 \)? What does this represent?
5. How can we interpret the meaning of the marginal cost formula in terms of production efficiency?

**Tip**: The marginal cost provides insight into how costs change with each additional unit, helping businesses make efficient production decisions.

What is the actual cost incurred in producing the 981st and the 1801st disc? What is the marginal cost when x = 980 and x = 1800?

Learn how to calculate the actual and marginal cost incurred in producing compact discs using the cost function C(x) = 2000 + 2x - 0.0001x^2. This example applies differentiation to find the marginal cost at specific production levels. Suitable for high school students in Grades 10-12.

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